Wilson loops in Large Symmetric Representations through a Double-Scaling Limit

We derive exact formulas for circular Wilson loops in the $\mathcal{N}=4$ and $\mathcal{N}=2^{\ast }$ theories with gauge groups $U(N)$ and $SU(N)$ in the $k$-fold symmetrized product representation. The formulas apply in the limit of large $k$ and small Yang-Mills coupling $g$, with fixed effective coupling $\kappa \equiv g^{2}k$, and for any finite $N$. In the $SU(2)$ and $U(2)$ cases, closed analytic formulas are obtained for any $k$, while the $1\mathrm{/}k$ series expansions are asymptotic. In the $N\gg 1$ limit, with $N\ll k$, there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the $k$-symmetric Wilson loops and chiral primary operators are also given.